Constraints on cosmic curvature with lensing time delays and gravitational waves

Abstract

Assuming the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ model, the CMB and BAO observations indicate a very flat Universe. Model-independent measurements are therefore worth studying. Time delays measured in lensed quasars provide the time delay distances. When compared with the luminosity distances from the supernova Ia (SNe Ia) observation, the measurements can provide the curvature information under the Distance Sum Rule of the Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) metric. This method is limited by the low redshifts of SNe Ia. In this work, we propose that gravitational waves from the Einstein Telescope, as standard sirens which reach higher redshifts covering the redshift range of lensed quasars from the Large Synoptic Survey Telescope, could provide much more stringent constraints on the curvature. We first consider a conservative case where only 100 gravitational waves with electromagnetic counterparts are available; the $1\ensuremath{\sigma}$ uncertainty for the curvature parameter ${\mathrm{\ensuremath{\Omega}}}_{k}$ is 0.057. In an optimistic case with 1000 signals available, then ${\mathrm{\ensuremath{\Omega}}}_{k}$ uncertainty is 0.027. Combining with SNe Ia from the Dark Energy Survey, ${\mathrm{\ensuremath{\Omega}}}_{k}$ can be further constrained to 0.027 and 0.018, respectively.